There are by now (at least) three different rigorous constructions of the quantum representations of \(\Gamma _g\) . The first uses representation theory of the quantum group \(U_q(\mathfrak {sl}(2))\) , with \(q\)  a fixed root of unity [1]}, [2]}, [3]}, the second gives a combinatorial construction using skein theory [4]}, and the third provides a geometric description involving the parallel transport of a projectively flat connection in a vector bundle over the moduli space of smooth compact Riemann surfaces of genus \(g\) .
The fact that these three constructions are equivalent follows from [5]} and [6]}.
